Constraint on the generalized Chaplygin gas as an unified dark fluid model after Planck 2015

Physics of the Dark Universe 22 (2018) 60–66

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Constraint on the generalized Chaplygin gas as an unified dark fluid model after Planck 2015 ∗

Hang Li a , , Weiqiang Yang b , Yabo Wu b a b

College of Medical Laboratory, Dalian Medical University, Dalian, 116044, PR China Department of Physics, Liaoning Normal University, Dalian, 116029, PR China

article

info

Article history: Received 20 April 2018 Received in revised form 8 July 2018 Accepted 6 September 2018 Keywords: Unified dark energy and dark matter model Cosmological constraint

a b s t r a c t The generalized Chaplygin gas could be considered as the unified dark fluid model because it might describe the past decelerating matter dominated era and at present time it provides an accelerating expansion of the Universe. In this paper, we employed the Planck 2015 cosmic microwave background anisotropy, type-Ia supernovae, observed Hubble parameter data sets to measure the full parameter space of the generalized Chaplygin gas as an unified dark matter and dark energy model. The model parameters Bs and α determine the evolutional history of this unified dark fluid model by influencing the energy density ρGCG = ρGCG0 [Bs + (1 − Bs )a−3(1+α ) ]1/(1+α ) . We assume the pure adiabatic perturbation of unified generalized Chaplygin gas. In the light of Markov Chain Monte Carlo method, we found that 0.020+0.051 +0.0208+0.1087 Bs = 0.759+ −0.032−0.046 and α = 0.0801−0.0801−0.0801 at 2σ level. The model parameter α is very close to zero, the nature of GCG model is very similar to cosmological standard model ΛCDM. © 2018 Elsevier B.V. All rights reserved.

1. Introduction In modern cosmology, Many theoretical models have been used to explain the current accelerating expansion [1]. Accelerating expansion of the Universe has been shown from the type Ia supernova (SNIa) observations [2,3] in 1998. During these years from that time, some other and updated observational results, including current Cosmic Microwave Background (CMB) anisotropy measurement from Planck 2015 [4–6], and the updated SNIa data sets from the Joint Light-curve Analysis (JLA) sample [7], also strongly support the present acceleration of the Universe. The latest release of Planck 2015 full-sky maps about the CMB anisotropies [6] indicates that baryon matter component is about 4% for total energy density, and about 96% energy density in the Universe is invisible which includes dark energy and dark matter. Considering the fourdimensional standard cosmology, this accelerated expansion for universe predict that dark energy (DE) as an exotic component with negative pressure is filled in the Universe. And it is shown that DE takes up about two-thirds of the total energy density from cosmic observations. The remaining one third is dark matter (DM). In theory, amount of DE models have already been constructed, for the reviews and papers please see [1,8–17]. However there exists another possibility that the invisible energy component is a unified dark fluid. i.e. a mixture of dark matter and dark energy. If one treats the dark energy and dark matter as an unified dark fluid, the corresponding models have been put forward and ∗ Corresponding author. E-mail address: [email protected] (H. Li). https://doi.org/10.1016/j.dark.2018.09.001 2212-6864/© 2018 Elsevier B.V. All rights reserved.

studied in Refs. [18–32]. In these unified dark fluid models, the Chaplygin gas (CG) and its generalized model have been widely studied in order to explain the accelerating universe [21–28]. The most interesting property for this scenario is that, two unknown dark sections—dark energy and dark matter can be unified by using an exotic equation of state. The original Chaplygin gas model can be obtained from the string Nambu–Goto action in the light cone coordinate [33]. For generalized Chaplygin gas (GCG), it emerges as an effective fluid of a generalized dbrane in a (d + 1, 1) space time, and its action can be written as a generalized Born–Infeld form [23]. Considering that the application of string theory in principle is in very high energy when the quantum effects is important in early universe [33]. The generalized Chaplygin gas (GCG) model is characterized by two model parameters Bs and α , which could be determined by the cosmic observational data sets. In order to constrain the model parameter space of GCG model, Xu [26] treated the dark energy and dark matter as a whole energy component, performed a global fitting on GCG model by the Markov Chain Monte Carlo (MCMC) method by the observational data sets CMB from WMAPseven-year [34], BAO [35], SNIa from Union2 [36] data. The tight 0.000970+0.00268 constraint had been obtained: α = 0.00126+ −0.00126−0.00126 and +0.0161+0.0307 Bs = 0.775−0.0161−0.0338 at 2σ level. For the very small values of GCG parameter α , it was concluded that GCG is very close to ΛCDM model. Amendola et al. [15] adopted the WMAP-first-year temperature power spectra [37] and SNIa data [2,3] to test the GCG model parameter space, and it was also concluded that GCG is very close to ΛCDM model. So in the light of previous reference, we will test the parameter space of GCG model with the recently released data sets, CMB from Planck 2015 [4–6], SNIa from JLA sample [7],

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and the observed Hubble parameter data [38], it is worthwhile to anticipate that a different constraint will be obtained. In this paper, the outline is as follows. In Section 2, we would show the background and perturbation equations of GCG model when the pure adiabatic contribution has been considered. In Section 3, based on the MCMC method, the global fitting results of GCG model parameters would be obtained by the joint observational data sets. Then, we might make some analysis on the measurement results. The conclusion would be drawn in the last section.

where the dot denotes the derivative of conformal time, the notations follow Ma and Bertschinger [40]. In our calculation, the adiabatic initial conditions are used.

2. The background and perturbation equations of generalized Chaplygin gas model

• CMB: We use CMB data from the Planck 2015 measurements

The GCG fluid is treated as an unified component in the Universe, its equation of state reads α pGCG = −A/ρGCG

(1)

where A and α are model parameters. By adopting the continuity equation, one could calculate the energy density of GCG fluid as

] 1 [ ρGCG = ρGCG0 Bs + (1 − Bs )a−3(1+α) 1+α

(2)

where Bs = A/ρ and α are the model parameters which could be constrained by the observational data sets. The parameter condition 0 ≤ Bs ≤ 1 is required to keep the positive energy density. When α = 0 in Eq. (2), we easily get the cosmological standard model ΛCDM; if α = 1 the CG model might be obtained. The equation of state of GCG is 1+α GCG0

w=−

Bs Bs + (1 − Bs )a−3(1+α )

.

(3)

H02

H =

{

(1 − Ωb − Ωr ) Bs + (1 − Bs )a

+Ωb a−3 + Ωr a−4

[

−3(1+α )

1 ] 1+α

} (4)

where H and H0 are the Hubble parameter and its present value, Ωb and Ωr are dimensionless energy density parameters of baryon and radiation. In Ref. [21], the author firstly studied the perturbation evolution of GCG fluid in order to explore the effects on the CMB anisotropic power spectra, sand then in Ref. [26], the author made a similar perturbation analysis by the assumption of pure adiabatic contribution. Under the pure adiabatic perturbation mode, the sound speed of GCG is cs2 =

δp p˙ = = −αw, δρ ρ˙

(5)

due to the non-positivity of equation of state w , α ≥ 0 is required to keep the non-negativity of sound speed, and the positive α is necessary for the stability of GCG perturbations [26], and the reasonable range is 0 ≤ α ≤ 1 from the detailed analysis in Refs. [21,39], when α is negative, the GCG model would possibly undergo catastrophic instabilities due to an imaginary speed of sound. According to the conservation of energy–momentum tensor µ Tν;µ = 0, ignoring the shear perturbation, one could the deduce the perturbation equations of density contrast and velocity divergence for GCG [26] h˙ δ˙GCG = −(1 + w)(θGCG + ) − 3H(cs2 − w)δGCG 2

θ˙GCG = −H(1 − 3cs2 )θGCG +

cs2 1+w

k2 δGCG

In this section we first describe the astronomical data with the statistical technique to constrain the GCG scenarios and the results of the analyses. We include the following sets of astronomical data. [4,5], where we combine the full likelihoods ClTT , ClEE , ClTE in addition with low−l polarization ClTE + ClEE + ClBB , which notationally is same with ‘‘PlanckTT, TE, EE + lowP’’ of Ref. [5]. • JLA: This is the Supernovae Type Ia sample that contains 740 data points spread in the redshift interval z ∈ [0.01, 1.30] [7]. This low redshifts sample is the first indication for an accelerating universe. • Cosmic Chronometers (CC): The Hubble parameter measurements from most old and passively evolving galaxies, known as cosmic chronometers (CC) have been considered to be potential candidates to probe the nature of dark energy due to their model-independent measurements. For a detailed description on how one can measure the Hubble parameter values at different redshifts through this CC approach, and its usefulness, we refer to [38]. Here, we use 30 measurements of the Hubble parameter at different redshifts within the range 0 < z < 2. So the total likelihood χ 2 can be constructed as 2 2 2 χ 2 = χCMB + χJLA + χCC .

where w is non-positive from the above equation. In the flat Universe, one has the Friedmann equation 2

3. Observational data sets and methodology

(6) (7)

(8)

In order to extract the observational constraints of the GCG scenarios, we use the publicly available Monte Carlo Markov Chain (MCMC) package COSMOMC [41] equipped with a convergence diagnostic followed by the Gelman and Rubin statistics, which includes the CAMB code [42] to calculate the CMB power spectra. We modified this code for the GCG model with the perturbation of unified dark fluid. We have used the following 7-dimensional parameter space P ≡ {ωb , 100θMC , τ , α, Bs , ns , log[1010 As ]}

(9)

where Ωb h2 stands for the density of the baryons and dark matter, 100θMC refers to the ratio of sound horizon and angular diameter distance, τ indicates the optical depth, α and Bs are two added parameters of GCG model, ns is the scalar spectral index, and As represents the amplitude of the initial power spectra. The pivot scale of the initial scalar power spectra ks0 = 0.05 Mpc−1 is used. A positive parameter α is required to keep the non-negativity of sound speed, and the positive α is necessary for the stability of GCG perturbations according to the analysis of Ref. [26], thus, α ≥ 0 is a compulsory condition for the observational constraint of GCG model. During the MCMC analysis, we generally fix some priors on the model parameters. Here, we show the priors set on various cosmological parameters, we take the following priors to model parameters: Ωb h2 ∈ [0.005, 0.1], ΘS ∈ [0.5, 10], τ ∈ [0.01, 0.8], α ∈ [0, 1], Bs ∈ [0, 1], ns ∈ [0.5, 1.5] and log[1010 As ] ∈ [2.7, 4]. 4. Analysis on the fitting results Let us summarize the main observational results extracted from the GCG unified model by using the three different combined observational data, CMB+CC, CMB+JLA, CMB+JLA+CC, described in the above section. In Table 1 we summarize the main results of global fitting results, at the first sight, the combination CMB+CC

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Fig. 1. The one-dimensional marginalized distribution on individual parameters, whose results are obtained by the different combination of observational data sets CMB + CC (gray line), CMB + JLA (red line), and CMB + JLA + CC (green line).

provide a slightly different constraint results of the model parameters, comparing with CMB+JLA and CMB+JLA+CC. One could see that for the combination CMB+CC, both the mean value and bestfit value of the parameter is negative, meanwhile the other two combination CMB+JLA and CMB+JLA+CC show a similarly positive value. All the results of three different observational combination support that the parameter α is very close to zero, its order is 10−2 from CMB+JLA and CMB+JLA+CC, it means that the nature of GCG model is very similar to cosmological standard model ΛCDM. In order to clearly and visually show the difference between different combination, we also plot the one-dimensional marginalized posterior distribution on individual parameters which are fully corresponding to Table 1. It is easily to see that for the parameters Ωb h2 , τ , log[1010 As ], ns are almost the same for the three different combination of data sets, however one would see the obvious difference about the mean values with error bars for the parameters 100θMC , Bs , α (H0 and Age/Gyr are the derived parameters). Besides, the combination CMB+JLA and CMB+JLA+CC provide the nearly identical results, and these two cases’ global fittings show us the tighter constraint than CMB+CC, to some extent, for the GCG model, the CC data set could not improve the constraint results. Then, we also show the one-dimensional marginalized posterior distribution in Fig. 1 and two-dimensional marginalized posterior distribution contours which include the GCG model parameters Bs and α are shown in Fig. 2, one could see that the parameter Ωb h2 owns the vague correlation with the other parameters,

the GCG model parameters Bs and α show the obvious positive correlation, both Bs and α show the distinct positive (negative) correlation with the derived parameter H0 (Age/Gyr). Here we also extract the two-dimensional contours for several parameters pair Bs − α , H0 − α , Age/Gyr − α , and the three-dimensional figures with colors between the parameters pair Bs − α and the present Hubble parameter H0 , which are shown in Figs. 3 and 4. The recent observational data sets support the tiny positive values of the parameter α for the three combinations CMB + CC , CMB + JLA and CMB + JLA + CC , the mean value of α of CMB + CC is slightly bigger than one of α of CMB+JLA and CMB+JLA+CC , α = 0 could be recovered at 1σ level for all the three observational combinations, the GCG model do not show the obvious deviation from the ΛCDM cosmology, in other words, the nature of GCG model is very similar to cosmological standard model ΛCDM. In order well understand the effects of model parameters on the CMB anisotropic power spectra, we plot the power spectra of CMB, where one of two model parameters α and Bs varies, where the other relevant parameters are fixed to their mean values as listed in Table 1. The panels of Fig. 5 show the effect of parameter α and Bs to CMB power spectra respectively. The model parameters α modifies the power law of the energy density of GCG, then it makes the gravity potential evolution at late epoch of the universe. As results, one can see Integrated Sachs–Wolfe (ISW) effect on the large scale as shown in the left panel of Fig. 5. In the early epoch, GCG behaves

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Fig. 2. The one-dimensional marginalized distribution on individual parameters and two-dimensional contours of GCG model at 1σ and 2σ regions, whose results are obtained by the different combination of observational data sets CMB + CC (gray line or contour), CMB + JLA (red line or contour), and CMB + JLA + CC (green line or contour).

like cold dark matter with almost zero equation of state and speed of sound cs2 , therefore the variation of the values of α will change the ratio of energy densities of the effective cold dark matter and baryons. One can read the corresponding effects from the variation of the first and the second peaks of CMB power spectra. For varied values of the model parameter Bs , decreasing the values of Bs , which is equivalent to increase the value of effective dimensionless energy density of cold dark matter, will make the equality of matter and radiation earlier, therefore the sound horizon is decreased. As a result, the first peak is depressed. The Fig. 6 shows CMB power spectra with mean values listed in Table 1 for GCG model and ΛCDM model, and the black dashed line is for ΛCDM model with mean values taken from Planck 2015 [6] constraint results. One

can see that GCG can match observational data points and ΛCDM model well. 5. Summary In this paper, we employed the Planck 2015 cosmic microwave background anisotropy, type-Ia supernovae, observed Hubble parameter data sets to measure the full parameter space of the generalized Chaplygin gas as an unified dark matter and dark energy model. The parameters Bs and α determine the evolutional history of this unified dark fluid model by influencing the energy density ρGCG = ρGCG0 [Bs + (1 − Bs )a−3(1+α) ]1/(1+α) . We assume the pure adiabatic perturbation of unified generalized Chaplygin gas, the positive α is necessary for the stability of GCG perturbations. In

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H. Li et al. / Physics of the Dark Universe 22 (2018) 60–66

Fig. 3. The two-dimensional contours of GCG model at 1σ and 2σ regions for the parameters pair, the dashed vertical line is corresponding to the parameter α = 0, whose results are obtained by the different combination of observational data sets CMB + CC (gray contour), CMB + JLA (red contour), and CMB + JLA + CC (green contour).

Fig. 4. The three-dimensional figures with colors of GCG model between the parameters pair Bs , α and derived parameter H0 for three different observational combination, the dashed vertical line is corresponding to the parameter α = 0, whose results are obtained by the different combination of observational data sets CMB + CC (gray contour), CMB + JLA (red contour), and CMB + JLA + CC (green contour).

Fig. 5. The effects on CMB temperature power spectra for the different values of the model parameter α and Bs . In the left figure, the red solid, green dotted–dashed, blue dashed lines are for α = 0.0801, 0.2, 0.5 (from top to bottom, on the left), respectively; in the right figure, the blue dashed, red solid, green dotted–dashed lines are for Bs = 0.859, 0.759, 0.659 (from top to bottom, on the right), respectively; the other relevant parameters are fixed with the mean values of CMB + JLA + CC as shown in Table 1.

the light of Markov Chain Monte Carlo method, we found that +0.020+0.051 +0.0208+0.1087 Bs = 0.759− 0.032−0.046 and α = 0.0801−0.0801−0.0801 at 2σ level. Tight constraint is obtained as shown in Table 1 and Fig. 2. All the results of three different observational combination support that the parameter α is very close to zero, its order is 10−2 from the combinations CMB+JLA and CMB+JLA+CC, it means that the nature of GCG model is very similar to cosmological standard model ΛCDM. The GCG model can match observational data points and ΛCDM model well. Currently available data sets of CMB, JLA and CC could not distinguish GCG model from ΛCDM model.

Besides, here, we do not use the baryon acoustic oscillation (BAO) data to test the GCG model parameter space, because if one wants to use the BAO data to test a unified dark fluid model, one needs to know the sound horizon at the redshift of drag epoch zd . Usually, zd is obtained by using the accurate fitting formula [43] which is valid if the matter scalings ρb ∝ a−3 and ρc ∝ a−3 are respected. Obviously, it is not true for the unified dark fluid model. One might reconsider the fiducial cosmology according to the analysis of [26,44,45], this would be an important research field in our future work.

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Table 1 The mean values of model parameters with 1σ and 2σ errors and best-fit values from the joint constraint, whose results are obtained by the different combination of observational data sets CMB + CC (2nd column), CMB + JLA (3rd column), and CMB + JLA + CC (4th column). Parameters

Mean ones from CMB + CC

Best fit

Mean ones from CMB + JLA

Best fit

Mean ones from CMB + JLA + CC

Ωb h2

0.00015+0.00030 0.02220+ −0.00015−0.00029

0.02223

+0.00016+0.00032 0.02220− 0.00016−0.00030

0.02226

0.00015+0.00030 0.02221+ −0.00015−0.00030

Best fit 0.02210

100θMC

0.00108+0.00152 1.02685+ −0.00068−0.00178

1.02756

+0.00070+0.00109 1.02729− 0.00046−0.00123

1.02766

0.00064+0.00105 1.02731+ −0.00048−0.00112

1.02684

τ

0.017+0.034 0.077+ −0.018−0.033

0.081

+0.017+0.033 0.077− 0.017−0.034

0.080

0.017+0.032 0.078+ −0.017−0.033

0.074

ns

0.0044+0.0089 0.9650+ −0.0044−0.0085

0.9646

+0.0046+0.0091 0.9651− 0.0047−0.0088

0.9682

0.0045+0.0089 0.9652+ −0.0045−0.0087

0.9649

ln(1010 As )

0.033+0.065 3.089+ −0.032−0.065

3.140

+0.034+0.064 3.090− 0.033−0.067

3.109

0.034+0.064 3.091+ −0.033−0.065

3.086

Bs

+0.031+0.086 0.783− 0.058−0.072

0.759

+0.020+0.057 0.760− 0.035−0.049

0.744

0.020+0.051 0.759+ −0.032−0.046

0.780

α

0.0351+0.2168 0.1421+ −0.1421−0.1421

0.1769

0.0184+0.1226 0.0822+ −0.0822−0.0822

0.0913

+0.0208+0.1087 0.0801− 0.0801−0.0801

0.0706

H0

1.38+3.97 69.76+ −2.48−3.31

68.69

0.97+2.52 68.74+ −1.43−2.26

68.36

0.92+2.32 68.73+ −1.36−2.09

69.39

Age/Gyr

0.054+0.085 13.768+ −0.041−0.092

13.782

+0.036+0.062 13.789− 0.032−0.070

13.792

0.034+0.060 13.788+ −0.031−0.064

13.784

of China under Grant No. 81601855, the Foundation of Education Department of Liaoning Province in China under Grant No. L2016036. W. Yang’s work is supported by the National Natural Science Foundation of China under Grants No. 11705079 and No. 11647153. Y. Wu’s work is supported by the National Natural Science Foundation of China under Grant No. 11575075. Appendix. The extended analysis for α < 0

Fig. 6. The CMB temperature power spectra v.s. multiple moment l, the red solid line is for the unified dark fluid model with mean values of CMB + JLA + CC as shown in Table 1, the black dashed line is for ΛCDM model with mean values from Planck 2015 analysis.

Acknowledgments The authors thank the referees for important comments. H. Li’s work is supported by the National Natural Science Foundation

In this paper, we have tested the generalized Chaplygin gas as an unified dark matter and dark energy model, with a prior of α ≥ 0, as is known from Refs. [21,39], the negative values of α would lead to catastrophic instabilities due to an imaginary speed of sound, here we would do some extended analysis of the negative α for the unified dark fluid model. In order to obtain the cosmological implication for α < 0, firstly we show the evolution of dimensionless energy density for the homogeneous background in the left panel of Fig. 7, for the mean value α = 0.0801 from CMB + JLA + CC and a tiny negative value of α = −5 × 10−5 , the background energy density evolution for these two cases are almost the same, however because of the imaginary speed of sound, a tiny negative value of α would strongly alter the CMB temperature power spectra, especially in the low−l region of the temperature power spectra, which could be clearly seen from the right panel of Fig. 7. And then, to follow the analysis of Ref. [39], we also show the power spectra of unified dark fluid for the tiny negative values of α , in Fig. 8, the catastrophic instabilities appear, the power spectra exhibit the blow-up increasing for the tiny negative value of α , which is

Fig. 7. The left panel is the evolution curves of dimensionless energy density Ωi (a) (the subscript b represents baryons) for the homogeneous background, the right panel is the CMB temperature power spectra, to compare the mean value of α from CMB + JLA + CC with the negative value of α , the other relevant parameters are fixed with the mean values of CMB + JLA + CC as shown in Table 1.

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Fig. 8. The power spectra of unified dark fluid for α = 0, α = −10−5 , α = −5 × 10−5 , the other relevant parameters are fixed with the mean values of CMB+JLA+CC as shown in Table 1.

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